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\title{Chapter 9. The Riemann Surface of An Algebraic Curve}
\author{GF ET AL}
%\date{2025年10月18日}

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% Section 9.7
%\section{9.0. } %%0
\begin{frame}[allowframebreaks]{9.0. intro.}

\vspace{-0.4cm}

In earlier chapters we were able to give \emph{rational parametrizations} of several curves $C \subset \mathbb{P}_2(\mathbb{C})$. 

A rational parametrization is a map
\[
\varphi: \mathbb{P}_1(\mathbb{C}) \to C
\]
that is bijective away from the singularities of $C$. 

In this last chapter we will show that such a parametrization always exists if we admit an arbitrary compact Riemann surface $S$ instead of $\mathbb{P}_1(\mathbb{C})$. 

This $S$ is uniquely determined by $C$, up to a biholomorphism; it can be called the \emph{resolution of singularities} of $C$. 

It leads to a better understanding of many properties of plane curves---the estimate of the number of singular points in Section~3.8, for instance, or the duality in Section~5.3. 

In particular, the Pl\"ucker formulas appear in a clearer light and a more general setting.

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% Section 9.1
\section{9.1. Riemann Surfaces} %%1
\begin{frame}[allowframebreaks]{9.1. }

\vspace{-0.4cm}

%\subsection*{9.1.}
First we briefly list the necessary facts about Riemann surfaces. For details and proofs, we refer the reader to the literature ([Fo2], for example).

\textbf{Definition.}
A Riemann surface is a connected Hausdorff topological space $S$, together with a \emph{complex atlas}
\[
(\psi_i : V_i \to U_i)_{i \in I}.
\]
This means that for all $i \in I$, the sets $V_i \subset C$, $U_i \subset S$ are open and the maps $\psi_i$ (the \emph{charts}) are homeomorphisms, and for all $i,j \in I$ the \emph{transition functions}
\[
\psi_{ij} : V_{ij} = \psi_i^{-1}(U_i \cap U_j) \to V_{ji} = \psi_j^{-1}(U_i \cap U_j)
\]
are biholomorphic.

A map
\[
\varphi : S \to T
\]
between Riemann surfaces is called \emph{holomorphic} if it is described by holomorphic functions when viewed in the charts.

A Riemann surface $S$ is a manifold of complex dimension one, so of real dimension two. Using the Cauchy-Riemann differential equations, one can see that the real Jacobian matrix of the $\psi_{ij}$ has positive determinant. Hence $S$ is orientable as a real surface.

It is known from topology that any compact orientable \emph{surface} (two-dimensional topological manifold) is homeomorphic to a sphere with $g$ handles. The number $g \in \mathbb{N}$ is called the \emph{genus} of the surface (see [M]).

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% Section 9.2
\section{9.2. Examples } %%2
\begin{frame}[allowframebreaks]{9.2. }

\vspace{-0.4cm}

%\subsection*{9.2.}
It is not hard to construct a complex atlas on a sphere with $g$ handles for any $g \in \mathbb{N}$. 

As a break from so much algebra, the procedure will be sketched here.

For $g = 0$, $S = \mathbb{P}_1(\mathbb{C})$ is the Riemann sphere with two charts, where
\[
U_0 = \mathbb{P}_1(\mathbb{C}) - \{(0:1)\}, \quad U_1 = \mathbb{P}_1(\mathbb{C}) - \{(1:0)\}.
\]

For $g \geq 1$, we start with a regular $4g$-gon $P \subset \mathbb{R}^2 = \mathbb{C}$ with edges
\[
a_1, b_1, a_1', b_1', \ldots, a_g, b_g, a_g', b_g',
\]
which are arranged and oriented as follows: On the boundary of $P$, if we identify the edges $a_i$ with $a_i'$ and $b_i$ with $b_i'$, oriented as in Figure 9.4, we obtain a surface $S$ of genus $g$. 

All the corners of $P$ become a point $o \in S$, and the edges $a_i$ and $b_i$ become closed paths $\alpha_i$ and $\beta_i$ based at $o$. 

The path
\[
\alpha_1\beta_1\alpha_1^{-1}\beta_1^{-1} \cdots \alpha_g\beta_g\alpha_g^{-1}\beta_g^{-1} \quad \text{in } S
\]
is null-homotopic because it corresponds to the boundary of $P$ and can be contracted through the interior of $P$. 

Now let
\[
U_0, U_1, \ldots, U_g, U_{g+1}, \ldots, U_{2g+1}
\]
be the open sets in $S$ defined by means of their preimages $\tilde{U}_i$ in $P$. 

Here $o$ is contained only in $U_0$, $U_i$ intersects only $\alpha_i$, $U_{g+i}$ intersects only $\beta_i$, and $U_{2g+1}$ intersects none of the paths. 

So in $P$ we have the following:
\begin{itemize}
\item $\tilde{U}_{2g+1}$ is connected,
\item $\tilde{U}_i$ has two connected components for $i = 1, \ldots, 2g$,
\item $\tilde{U}_0$ splits into $4g$ connected components.
\end{itemize}

Now we turn to the construction of the charts
\[
\psi_i : V_i \to U_i.
\]
For $i = 2g+1$, we can choose $V_{2g+1}$ to be $\tilde{U}_{2g+1}$ and $\psi_{2g+1}$ to be the restriction of the canonical map $P \to S$. 

For $i = 1, \ldots, 2g$, we choose the $V_i$ to be disks. 

If we map two halves to the two components of $\tilde{U}_i$, we obtain $\psi_i$. 

For $i = 0$ we divide a disk of suitable radius into $4g$ equal sectors. 

These have central angle $\varphi = \pi/2g$ and the components of $\tilde{U}_0$ have central angle $(2g - 1)\varphi$, so we can assemble $\psi_0$ from translations (and $2g - 1$st powers). 

The resulting transition functions are biholomorphic. 

We have proved the following result.

\textbf{Theorem.}
For every $g \in \mathbb{N}$, a compact, orientable surface of genus $g$ can be made into a Riemann surface.

The complex structure is unique only for $g = 0$ (see [Fo2], 16.13). 

The set of \emph{complex structures} (i.e.\ the biholomorphic equivalence classes of atlases) has one complex parameter for $g = 1$, and $3g - 3$ complex parameters for $g \geq 2$. 

The space of these parameters is called \emph{Teichmüller space}. 

With the construction above, at least we have determined one point in it.

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% Section 9.3
\section{9.3. Desingularization of An Algebraic Curve} %%3
\begin{frame}[allowframebreaks]{9.3. }

\vspace{-0.4cm}

%\subsection*{9.3.}
Riemann surfaces have complex dimension one; complex manifolds of dimension $n$ can be defined in exactly the same way. 

For $n = 2$, we construct an atlas on $\mathbb{P}_2(\mathbb{C})$ consisting of three charts. 

For $i = 0,1,2$, let
\[
U_i := \{(x_0:x_1:x_2) \in \mathbb{P}_2(\mathbb{C}) : x_i \neq 0\}
\]
and let $V_i = \mathbb{C}^2$. 

For $i = 0$, the chart $\psi_0$ is given by
\[
\psi_0: \mathbb{C}^2 \to U_0, \quad (y_1,y_2) \mapsto (1:y_1:y_2),
\]
and similarly for $i = 1,2$. 

For practice, the reader is invited to check that the resulting transition functions $\psi_{ij}$ are biholomorphic.

Thus, for a Riemann surface $S$, it is clear when a map
\[
\varphi: S \to \mathbb{P}_2(\mathbb{C})
\]
is holomorphic: one checks it in the local charts. 

If $C \subset \mathbb{P}_2(\mathbb{C})$ is an algebraic curve, a map
\[
\varphi: S \to C
\]
is called holomorphic if it is holomorphic as a map to $\mathbb{P}_2(\mathbb{C})$. 

With these preliminaries out of the way, we can state the theorem.

\textbf{Theorem.}
For every irreducible algebraic curve $C \subset \mathbb{P}_2(\mathbb{C})$, there exist a compact Riemann surface $S$ and a holomorphic map
\[
\varphi: S \to C
\]
with the following properties:
\begin{enumerate}[label=(\roman*)]
\item Let $C' := C - \mathrm{Sing}\,C$ be the smooth part of $C$, and let $S' := \varphi^{-1}(C') \subset S$. Then
\[
\varphi':= \varphi|_{S'}: S' \to C'
\]
is biholomorphic.
\item For every $p \in C$ there is a bijective map
\[
\varphi^{-1}(p) \to \{\text{branches of } C \text{ at } p\}.
\]
In particular, $\varphi^{-1}(p)$ is finite for every $p$.
\end{enumerate}

A Riemann surface $S$ with properties (i) and (ii) is uniquely determined up to biholomorphic equivalence. 

$\varphi: S \to C$ is called the \emph{resolution of singularities} of $C$.

\textbf{Consequence.}
For any irreducible curve $C \subset \mathbb{P}_2(\mathbb{C})$, we can define the genus
\[
g(C) := \text{genus of } S.
\]
$C$ is said to be \emph{rational} if $g(C) = 0$.

\textbf{Remark.}
The theorem produces a Riemann surface for every plane curve. 

Conversely, as a consequence of the Riemann-Roch theorem, every compact Riemann surface $S$ can be realized as a smooth algebraic curve $S \subset \mathbb{P}_3(\mathbb{C})$. 

If we choose a suitable point $z \in \mathbb{P}_3(\mathbb{C}) - S$ as center, we obtain a projection
\[
\pi: \mathbb{P}_3(\mathbb{C}) - \{z\} \to \mathbb{P}_2(\mathbb{C})
\]
such that
\[
\varphi|_S: S \to C := \pi(S)
\]
is biholomorphic almost everywhere (see [H], Ch.~IV). 

Thus every compact Riemann surface occurs as the desingularization of a plane algebraic curve $C$. 

The projection can even be chosen so that $C$ is a Pl\"ucker curve and has at most simple double points as singularities (see Section~5.7).

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% Section 9.4
\section{9.4.} %%4
\begin{frame}[allowframebreaks]{9.4. }

\vspace{-0.4cm}

%\subsection*{9.4.}
For the \emph{proof of Theorem~9.3} we first construct a Riemann surface $S$. 

This is done by patching together a large number of open sets $V \subset C$.

If $p \in C'$, then by the implicit function theorem (Section~6.9) there exist an open set $V_p \subset C$, a neighborhood $p \in W_p \subset \mathbb{P}_2$, and a biholomorphic map
\[
\psi_p : V_p \to C \cap W_p \subset C'.
\]

For each $q \in \mathrm{Sing}\,C$, we choose a neighborhood $q \in W_q \subset \mathbb{P}_2$ such that the $W_q$ are pairwise disjoint and
\[
C \cap W_q = C_{q,1} \cup \cdots \cup C_{q,k_q},
\]
where the $C_{q,i}$ are representatives of the branches of $C$ at $q$ (see Section~6.14). 

In particular, let
\[
C_{q,i} \cap C_{q,j} = \{q\} \quad \text{for } i \neq j.
\]

Further, let $W_q$ be chosen so small that for every $i$ there is a Puiseux parametrization
\[
\psi_{q,i} : V_{q,i} \to C_{q,i},
\]
where $V_{q,i} \subset \mathbb{C}$ is open. 

Now we take the \emph{disjoint union} of all the open subsets in $\mathbb{C}$ obtained above:
\[
M := \bigcup_{p \in C'} V_p \cup \bigcup_{q \in \mathrm{Sing}\,C} V_{q,1} \cup \cdots \cup V_{q,k_q}.
\]

The maps $\psi_p$ and $\psi_{q,i}$ together yield a holomorphic map
\[
\psi : M \to C.
\]

Now things are patched together in $M$ as follows: for $p,p' \in C'$ and $q \in \mathrm{Sing}\,C$, we have
\begin{align*}
v \in V_p \text{ and } v' \in V_{p'} \text{ are equivalent} &\iff \psi_p(v) = \psi_{p'}(v') \in C', \\
v \in V_p \text{ and } v' \in V_{q,i} \text{ are equivalent} &\iff \psi_p(v) = \psi_{q,i}(v') \in C'.
\end{align*}

There is no patching between the sets $V_{q,i}$ and $V_{q,j}$. 

Let $S$ denote the quotient of $M$ by the equivalence relation given above, endowed with the quotient topology. 

The map $\psi$ induces a map
\[
\varphi : S \to C.
\]

We must show that it has all the properties stated in the theorem.


(a) $\varphi' : S' \to C'$ is bijective: This follows from the construction.

(b) $S$ is a Hausdorff space: The proof must be broken into cases and is left to the reader.

(c) $S$ is compact: Let
\[
S = \bigcup_{i \in I} U_i, \quad \text{where } U_i \subset S \text{ are open}.
\]

We use the compactness of $C$, but have to work a bit harder because the sets $\varphi(U_i) \subset C$ need not be open. 

Since the exceptional set $A := \varphi^{-1}(\mathrm{Sing}\,C) \subset S$ is finite, it is compact. 

We define
\[
I_0 := \{ i \in I : U_i \cap A \neq \emptyset \}, \quad I_1 := I - I_0.
\]

Since
\[
A \subset \bigcup_{i \in I_0} U_i,
\]
there exists a finite subset $J_0 \subset I_0$ such that
\[
A \subset \bigcup_{i \in J_0} U_i =: U.
\]

$A$ is finite, so $\varphi(U)$ is open. 

We define $U_i' := U_i - A$ for $i \in I_0 - J_0$. 

This gives the open cover
\[
C = \varphi(U) \cup \bigcup_{i \in I_0 - J_0} \varphi(U_i') \cup \bigcup_{i \in I_1} \varphi(U_i).
\]

Since $C$ is compact, there exist finite subsets $I_0' \subset I_0 - J_0$ and $J_1 \subset I_1$ such that
\[
C = \varphi(U) \cup \bigcup_{i \in I_0'} \varphi(U_i') \cup \bigcup_{i \in J_1} \varphi(U_i).
\]

Setting $I^* := J_0 \cup I_0' \cup J_1 \subset I$ gives
\[
S = \bigcup_{i \in I^*} U_i.
\]

(d) $S$ is connected: Here we use the irreducibility of $C$. 

More generally, it can be shown that to every irreducible component of $C$ there corresponds a connected component of $S$.



Let $C = V(F)$, where $F \in \mathbb{C}[X_0,X_1,X_2]$ is a homogeneous, irreducible polynomial of degree $n \geq 1$. 

We may assume that the point $q = (0:0:1) \notin C$. 

Then, up to a factor in $\mathbb{C}^*$,
\[
F = X_2^n + A_1X_2^{n-1} + \cdots + A_n, \quad \text{where } A_i \in \mathbb{C}[X_0,X_1],\ \deg A_i = i.
\]

Consider the maps
\[
S \xrightarrow{\varphi} C \xrightarrow{\pi} \mathbb{P}_1(\mathbb{C}),
\]
where $\pi$ denotes the projection with center $q$. 

Let $\Delta_F \in \mathbb{C}[X_0,X_1]$ be the discriminant of $F$ (see Section~A.1.2). 

$\Delta_F$ is homogeneous by Theorem~A.1.3, so we can define the exceptional set
\[
B := V(\Delta_F) \subset \mathbb{P}_1(\mathbb{C}).
\]

Since $F$ is irreducible, $\Delta_F \neq 0$; hence $B$ is finite. 

Let
\[
C^* := C - \pi^{-1}(B), \quad \pi^* = \pi|_{C^*} : C^* \to \mathbb{P}_1(\mathbb{C}) - B.
\]

Now it is easy to see that $C^* \subset C'$ and that $\pi^*$ is an $n$-fold covering (see Section~7.9). 

Suppose there is a connected component $T \subset S$ with $\emptyset \neq T \neq S$. 

Since the exceptional set
\[
\widetilde{B} := \varphi^{-1}(\pi^{-1}(B)) \subset S
\]
is finite, $T^* := T - \widetilde{B}$ is connected. 

Hence
\[
D := \varphi(T^*) \subset C^*
\]
is connected. 

By Remark~2 of Section~A.2.1,
\[
\eta := \pi|_D : D \to \mathbb{P}_1(\mathbb{C}) - B
\]
is itself a covering. 

Let $m$ be the sheet number of $\eta$. 

Since $\varphi'$ is bijective, it follows that $0 < m < n$.

It will now be shown that $F$ must have a factor $G$ of degree $m$. 

To make the construction of $G$ clear, we first modify the given $F$ slightly. 

Consider the two embeddings
\[
\iota: \mathbb{C} \times \mathbb{C} \to \mathbb{P}_2(\mathbb{C}), \quad (y_1,y_2) \mapsto (1:y_1:y_2);
\]
\[
\tilde{\iota}: \mathbb{C} \times \mathbb{C} \to \mathbb{P}_2(\mathbb{C}), \quad (z_0,z_2) \mapsto (z_0:1:z_2).
\]

We have the corresponding transformations
\[
Y_1 = \frac{X_1}{X_0}, \quad Y_2 = \frac{X_2}{X_0} \quad \text{and} \quad Z_0 = \frac{X_0}{X_1}, \quad Z_2 = \frac{X_2}{X_1}.
\]


From $F$ we obtain the polynomials
\[
f(Y_1,Y_2) := F(1,Y_1,Y_2) = \sum_{i=0}^n a_i Y_2^{n-i},
\]
\[
\tilde{f}(Z_0,Z_2) := F(Z_0,1,Z_2) = \sum_{i=0}^n \bar{a}_i Z_2^{n-i},
\]
where
\[
a_i \in \mathbb{C}[Y_1], \quad \bar{a}_i \in \mathbb{C}[Z_0], \quad \deg a_i,\ \deg \bar{a}_i \leq i, \quad \text{and}
\]
\[
X_0^i a_i\left(\frac{X_1}{X_0}\right) = X_1^i \bar{a}_i\left(\frac{X_0}{X_1}\right) = A_i(X_0,X_1). \tag{*}
\]

For the affine parts of $C$, we have
\[
C_0 = \iota^{-1}(C) = V(f) \quad \text{and} \quad \widetilde{C}_0 := \tilde{\iota}^{-1}(C) = V(\tilde{f}).
\]

The projection $\pi$ with center $q$ is given in affine coordinates by
\[
\begin{array}{ccc}
\iota^{-1}(C^*) & = & C_0^* \subset \mathbb{C} \times \mathbb{C}, \quad (y_1,y_2) \\
& \downarrow & \downarrow \\
& \pi_0^* & \quad \downarrow \\
& \mathbb{C} - B_0 & \subset \mathbb{C}, \quad y_1.
\end{array}
\]

Here $B_0$ is the affine part of the exceptional set $B$. 

Since the map $\pi_0^*$ is a restriction of $\pi^*$, it is itself a covering. 

Hence for every point $p \in \mathbb{C} - B_0$ there exists a neighborhood $W$ and bounded holomorphic functions $\psi_i \in \mathcal{O}(W)$ such that
\[
f(Y_1,Y_2) = \prod_{i=1}^n \bigl(Y_2 - \psi_i(Y_1)\bigr) \quad \text{in } W \times \mathbb{C}.
\]

The coefficients of $f$ are the elementary symmetric functions:
\[
a_i = s_i(\psi_1,\ldots,\psi_n) \quad \text{in } W.
\]

Similarly, we obtain
\[
\tilde{f}(Z_0,Z_2) = \prod_{i=1}^n \bigl(Z_2 - \tilde{\psi}_i(Z_0)\bigr), \quad \bar{a}_i = s_i(\tilde{\psi}_1,\ldots,\tilde{\psi}_n),
\]
in the $Z$-coordinates. 

Now, reversing this computation, we can start with the connected component $D \subset C^*$ and produce a factor $G$ of $F$. 

To do this, for some $p \in \mathbb{C} - B_0$, let the $\psi_i$ be numbered so that
\[
(x,\psi_i(x)) \in D \quad \text{for } i = 1,\ldots,m \text{ and } x \in W.
\]


We then define
\[
g(Y_1,Y_2) := \prod_{i=1}^m (Y_2 - \psi_i(Y_1)) \quad \text{in } W \times \mathbb{C}.
\]

The coefficients of $g$ are given by the elementary symmetric functions $t_1,\ldots,t_m$ in $m$ variables:
\[
g = Y_2^m + b_1(Y_1)Y_2^{m-1} + \cdots + b_m(Y_1), \quad b_j = t_j(\psi_1,\ldots,\psi_m).
\]

By the symmetry of the $t_j$, we thus obtain functions $b_j$ that are holomorphic in $\mathbb{C} - B_0$. 

Since $q \notin C$, the $\psi_i$ are still bounded in $B_0$, so the $b_j$ can be extended holomorphically to $\mathbb{C}$.

We claim that the $b_j$ are actually polynomials. 

To show this, we exploit the fact that, proceeding as above, we can obtain a function
\[
\bar{g}(Z_0,Z_2) = Z_2^m + \bar{b}_1(Z_0)Z_2^{m-1} + \cdots + \bar{b}_m(Z_0)
\]
in the $Z$-coordinates, with holomorphic $\bar{b}_j$. 

By construction, we have an analogue of (*):
\[
X_0^j b_j\left(\frac{X_1}{X_0}\right) = X_1^j \bar{b}_j\left(\frac{X_0}{X_1}\right). \tag{**}
\]

This means that at the point $\infty$, where $X_0 = 0$, each entire function $b_j$ has at most a pole of order $j$. 

Hence the $b_j$ are polynomials of degree $\leq j$. 

Thus
\[
B_j(X_0,X_1) := X_0^j b_j\left(\frac{X_1}{X_0}\right) \in \mathbb{C}[X_0,X_1]
\]
is homogeneous of degree $j$, and by construction
\[
G := X_2^m + B_1 X_2^{m-1} + \cdots + B_m
\]
is a proper divisor of $F$. 

But this contradicts the irreducibility of $F$. 

We have proved assertion (d).

The rest of the proof is routine and recommended to the reader as an exercise. 

To prove the uniqueness of $S$, use the Riemann extension theorem for holomorphic functions.
%\qed

\end{frame}
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% Section 9.5
\section{9.5.} %%5
\begin{frame}[allowframebreaks]{9.5. }

\vspace{-0.4cm}

\end{frame}
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% Section 9.6
\section{9.6.} %%6
\begin{frame}[allowframebreaks]{9.6. }

\vspace{-0.4cm}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 9.7
\section{9.7. The genus formula for smooth curves} %%1
\begin{frame}[allowframebreaks]{9.7. }

\vspace{-0.4cm}

%\subsection*{9.7.} 
The Riemann-Hurwitz formula can be used to compute the genus of an irreducible algebraic curve $C \subset \mathbb{P}_2(\mathbb{C})$. 

To do this, we use the maps
\[
S \xrightarrow{\varphi} C \xrightarrow{\pi} \mathbb{P}_1(\mathbb{C}),
\]
where $\varphi$ is the resolution of singularities of Section 9.3 and $\pi$ is a projection with center $z$ off $C$. 

Then
\[
\chi := \pi \circ \varphi: S \to \mathbb{P}_1(\mathbb{C})
\]
is a branched covering with sheet number $n = \deg C$. 

It remains to compute the branching order $\nu$ of $\chi$. 

This is especially easy when $C$ is smooth.

\textbf{Lemma.}
Let $C \subset \mathbb{P}_2(\mathbb{C})$ be a smooth, irreducible curve of degree $n$. 

Then its branching order satisfies
\[
\nu = n(n-1)
\]
if the center $z$ of the projection $\pi$ is chosen to be sufficiently general.

Setting $T = \mathbb{P}_1$ in the Riemann-Hurwitz formula gives an immediate corollary.

\textbf{Genus Formula.}
A smooth, irreducible curve $C \subset \mathbb{P}_2(\mathbb{C})$ of degree $n$ has genus
\[
g = \tfrac{1}{2}(n-1)(n-2).
\]

\textbf{Proof of the lemma.}
For a smooth curve we may assume that $S = C$, so what we have to determine is the branching order of the projection $\pi$. 

Let $z = (0:0:1)$ be the center of $\pi$. 

We assume that $z \notin C$. 

Then $\pi$ is given by
\[
C \to \mathbb{P}_1(\mathbb{C}), \quad p = (p_0:p_1:p_2) \mapsto q = (p_0:p_1).
\]

The geometry underlying the lemma is as follows: $p$ is a branch point of $\pi$ if and only if the line of projection $z \vee q$ is a tangent at $p$. 

If $z$ does not lie on any bitangent or inflectional tangent, then there are exactly $n^* = n(n-1)$ simple tangents from $z$ to $C$ (see Section~5.7). 

Being a simple tangent means $\nu_p(\pi) = 1$, and this gives the assertion.

Whoever finds this too short will have to put up with some computations. 

We write a minimal polynomial of $C$ in the form
\[
F = X_2^n + A_1X_2^{n-1} + \cdots + A_n, \quad \text{where } A_i \in \mathbb{C}[X_0,X_1].
\]
By Appendix~1.3, the discriminant $\Delta \in \mathbb{C}[X_0,X_1]$ has degree $n(n-1)$. 

We choose coordinates such that
\[
(0:1) \notin M := V(\Delta) \subset \mathbb{P}_1(\mathbb{C}).
\]
If we set $C' := C - \pi^{-1}(M)$, then the restriction
\[
\pi': C' \to \mathbb{P}_1(\mathbb{C}) - M
\]
is an unbranched covering. 

Thus we can look for all the branchings in the affine part
\[
\mathbb{C} \times \mathbb{C} \to \mathbb{P}_2(\mathbb{C}), \quad (y_1,y_2) \mapsto (1:y_1:y_2).
\]
If $f(Y_1,Y_2) = F(1,Y_1,Y_2)$ and $C_0 = V(f) \subset \mathbb{C} \times \mathbb{C}$, then
\[
\pi_0 = \pi|_{C_0}: C_0 \to \mathbb{C} \quad \text{is given by} \quad (y_1,y_2) \mapsto y_1.
\]

To simplify notation in describing the map $\pi$ at $p \in C_0$, we suppose $p = (0,0)$. 

Now we claim that
\[
\mathrm{ord}_p(\pi) = \mathrm{ord}\, f(0,Y_2). \tag{*}
\]
This follows from the Weierstrass preparation theorem (Section~6.7) and a Puiseux parametrization (Section~7.8): If $k := \mathrm{ord}\, f(0,Y_2)$, then we can substitute an irreducible Weierstrass polynomial of degree $k$ for $f$ in a neighborhood of $p$ (since $f$ is smooth, it is locally irreducible), and we can parametrize $C$ by
\[
s \mapsto (s^k, \varphi(s)).
\]
Composing this with $\pi$ gives the map $t = s^k$, which proves (*). 

The number $k$ has yet another meaning:
\[
k = \mathrm{mult}_p(C \cap L_q),
\]
where $L_q := z \vee q$ is the ``ray of projection'' through $p$. 

If the center $z$ is chosen off the bitangents and inflectional tangents of $C$, then $k \leq 2$. 

Hence the exceptional set $M$ consists of exactly $n(n-1)$ points, and over each one lies a branch point of order 1.



\end{frame}

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